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Some unconventional PDE problems from photonics

Robert Kohn (Courant Institute)

Location:  Hill 705
Date & time: Wednesday, 06 March 2024 at 3:30PM - 4:30PM

Speaker: Robert Kohn (Courant Institute)Title: Some unconventional PDE problems from photonicsAbstract: Electromagnetic waves are described by Maxwell's equations. For transverse-magnetic waves in a time-harmonic setting, the Maxwell system reduces to a simple-looking divergence-form Helmholtz equation: div( a(x,omega) grad u) + omega^2 m(x, omega) u = 0. Here u is scalar valued, omega is the frequency, and the coefficients a(x,omega) and m(x,omega) represent the material properties of the medium -- specifically, a(x,omega) is 1/(dielectric permittivity) and m(x,omega) is the magnetic permeability. One might think that we have, by now, a full understanding of the mathematical issues raised by this equation -- but in fact it isn't so. Indeed, due to the frequency-dependence of the permittivity, there are materials for which the dielectric permittivity is near zero or even negative at frequencies of practical interest. The analysis of devices made by combining such materials with ``ordinary'' dielectrics (for which the permittivity is positive) raises numerous PDE issues that are still not well understood.

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